XIV. A Method op Geometrical Representatton of the 

 Thermodynamic Properties of Substances by means op Sur- 

 faces. By J. WiLLARD GiBBS. 



The leading thermodynamic properties of a fluid are determined 

 l)y the relations which exist between the volume, pressure, tempera- 

 ture, energy, and entropy of a given mass of the fluid in a state of 

 thermodynamic equilibrium. The same is true of a solid in regard to 

 those properties which it exhibits in processes in which the pres- 

 sure is the same in every direction about any point of the solid. 

 But all the relations existing between these five quantities for any 

 substance (three independent relations) may be deduced from the 

 single relation existing for that substance between the volume, energy, 

 and entropy. This may be done by means of the general equation, 



de = t ch] — p dv, (1 )* 



where v, j), ^5 ^? '■^^^^ V denote severally the volume, pressure, absolute 

 temperature, energy, and entropy of the body considered. The sub- 

 script letter after the diflerential coefiicient indicates the quantity 

 which is supposed constant in the differentiation. 



Bepresentation of Volume, Entropy, Energy, Pressure, and Tem- 

 perature. 

 Now the relation between the volume, entropy, and energy may 

 be i-epresented by a surface, most simply if the rectangular co-ordin- 

 ates of the various points of the surface are made equal to the vol- 

 ume entropy, and energy of the body in its various states. It may 

 be interesting to examine the properties of such a surface, Avhich we 

 will call the thermodynamic surface of the body for wJiich it is 

 foi'med.f 



* For the demonstration of tliis equation, and in regard to the units used in the 

 measurement of the quantities, the reader is referred to page 310 of this vohune. 



•I- Professor J. Thomson has proposed and used a surface in which the co-ordinates 

 are proportional to the volume, pressure, and temperature of the body. (Proc. Roy. 

 Soc, Nov. 16, 1811, vol. XX, p. 1 ; and Phil. Mag., vol. xliii, p. 227). It is evident, 



